\begin{abstract} Let $(R,\fm)$ be a commutative Noetherian local ring and $M$ be a finitely generated $R$-module of dimension $d$. Then the following statements hold: \begin{itemize} \item[(a)] If $\width(H_{\fm}^i(M))\geq {i-1}$ for all $i$ with $2\leq i d$, then $H_{\fm}^d(M)$ is co-Cohen-Macaulay of Noetherian dimension $d$. \item[(b)] If $M$ is an unmixed $R$-module and $\depth{M}\geq{d-1}$, then $H_{\fm}^d(M)$ is co-Cohen-Macaulay of Noetherian dimension $d$ if and only if $H_{\fm}^{d-1}(M)$ is either zero or co-Cohen-Macaulay of Noetherian dimension ${d-2}$. \end{itemize} As consequence, if $H_{\fm}^i(M)$ is co-Cohen-Macaulay of Noetherian dimension $i$ for all $i$ with $0\leq i
